Covariate modeling

• When: Ideally after establishing the base model.
• Why: Help us understand which patient- or drug-specific characteristics are important determinants of PK parameters.
  • Mechanistic understanding:
    • Impact of disease type on CL.
    • Drug-drug interactions.
  • Controlling PK variability:
    • Dose selection for a special population (e.g., obese patients).
    • Dosing interval adjustment for extended release formulation.
• How: Specify a mathematical relationship between a PK parameter and covariates.
  • Usually requires adding parameters (mostly THETAs) in the model.

Continuous covariates

• Example: Body weight (WT) on CL
  • Linear:
    • CL=THETA(1)*(1+THETA(2)*(WT-70))
  • Piece-wise (hockey-stick) linear:
    • IF(WT.LE.50) CL=THETA(1)*(1+THETA(2)*(WT-70))
    • IF(WT.GT.50) CL=THETA(1)*(1+THETA(3)*(WT-70))
  • Exponential:
    • CL=THETA(1)*EXP(THETA(2)*(WT-70))
  • Power:
    • CL=THETA(1)*(WT/70)**THETA(2)
• Centering: in all of the above case, when WT=70, model collapse to CL=THETA(1)
  • Interpretation of THETA(1): Typical CL for a subject with 70 kg body weight.
  • If replacing 70 with 60, for example, the interpretation of THETA(1) changes to “typical CL for a subject with 60 kg body weight”, with no impact on model fittings.

Categorical covariates

• Example: Sex (Male=1; Female=0)
  • Linear:
    • CL=THETA(1)
    • IF(SEX.EQ.1) CL=THETA(1)*(1+THETA(2))

Covariate modeling approaches

Covariate modeling approaches

• Active area of interest.
• No single method is universally accepted as the “gold-standard”.
• Choice of method depends on modeling objectives and context (i.e., fit-for-purpose)

Covariate examinations: BEFORE you start covariate modeling

• Know you covariates
  • Missing values?
    • No missing values-Perfect!
    • Small proportion (5%-10%):
      • Impute with median/mode.
      • Multiple imputation with chain equations (MICEs).
    • Large proportion (e.g., 30%)
      • Treat “missing” as a separate category.
  • Range and distribution of continuous covariates (e.g., body weight).
    • Distribution wide enough?
  • Count in each category of categorical covariates (e.g., sex)
    • Imbalance across category? Male:Female=20:1

Data reduction: BEFORE you start covariate modeling

• Examine covariate distributions for correlation/colinearity
  • Select covariates carrying unique information (relatively independent).
  • Rule of thumb: be cautious when \(R^2>0.3\).
• Exclude or combine covariates carrying repetitive information.
  • WT, HT, BSA and BMI are highly correlated. If one included as a covariate, others may not need to be included.
  • BLACK (No=0, Yes=1), WHITE (No=0, Yes=1) and ASIAN (No=0, Yes=1). May lump into one covariate (BLACK=1, WHITE=2, ASIAN=3, OTHER=4).

Other consideration points

• Was the study designed to estimate a covariate effect?
  • SNP impact on CL.
• Inclusion criteria of a clinical study may impact the choice of covariates.
  • A study enroll only female patients (sex may be difficult to be explored as a covariate).
• Prior knowledge (very subjective).
  • Eye colors on CL?
• Clinical interest of covariates

Selection methods-(SCM)

Nesting example

• Model 1:

TVCL = THETA(1)

• Model 2:

TVCL = THETA(1)*(WT/70)**THETA(2)

• If WT/70 equals 1, then (WT/70)**THETA(2) equals 1, resulting in TVCL = THETA(1).
• So Model 1 is nested within Model 2.

Another nesting example

• Model 1:

TVCL = THETA(1)

• Model 2:

    TVCL = THETA(1)
    IF(SEX.EQ.0) TVCL = TVCL * THETA(2)

• With THETA(2) fixed to 1.0 (it’s null value)
  • TVCL = THETA(1) (if male)
  • TVCL = THETA(1) * 1 (if female)
• No predictive difference between males and females when THETA(2) is set to its null value
• Smaller model (without SEX) is nested

Nesting?

• Model 1:

TVCL = THETA(1)

• Model 2:

TVCL = THETA(1)*WT

• No biological meaningful WT reference value to remove the effect of WT.
• Be careful. You can add COV effect without nesting.

Statistical significance vs. clinical relevance

Example 1:

TVCL = THETA(1)
IF(SEX.EQ.0) TVCL = TVCL*THETA(2)

• Fix THETA(2)=1, OFV=69.9
• Estimate THETA(2), THETA(2)estimated as 1.1, OFV=62.1
• Clinical relevant?
  • Females have a 10% faster CL than males
• Statistical significant?
  • dOFV=69.9-62.1=7.8
  • Using a P value criteria (0.01) with 1 DF, if dOFV>6.63, the COV is considered statistically significant.
• Statistically significant \(\neq\) clinically relevant
  • Maybe you just happen to have a large sample size (n=3000).
  • Any small difference between subgroups can be detected.
  • Do we really care 10% faster CL clinically in females than males clinically.

Statistical significance vs. clinical relevance

Example 2:

TVCL = THETA(1)
IF(SEX.EQ.0) TVCL = TVCL*THETA(2)

• Fix THETA(2)=1, OFV=21.3
• Estimate THETA(2), THETA(2)estimated as 0.4, OFV=20.1
• Clinical relevant?
  • Females have a 60% slower CL than males
• Statistical significant?
  • dOFV=21.3-20.1=1.2
  • Using a P value criteria (0.01) with 1 DF, only if dOFV>6.63, the COV is considered statistically significant.
• Statistically insignificant \(\neq\) clinically irrelevant
  • Maybe your sample size is too small (n=8).
  • Although COV effect is large (60%), the study is not sufficiently powered to detect it

Pre-defined methods: FFEM

Full fixed-effect modeling (FFEM)[^3] procedure:

• Develop a stable base model.

• Thoughful consideration given to potential covariate-parameter relationships.

• Full covariate model is constructed and checked for goodness-of-fit.

• Check point estimates and 95% confidence intervals (CIs).

  • Unimportant covariates are driven to 0 during estimation with wide CIs.

• If the inclusion of an unimportant covariate makes a model “unstable”, consider exclude it from the model.

  • Model cannot converge

  • Covariance step fails

  • Very wide and unacceptable CIs (e.g., 2000% relative standard error)

• Make inference based on simulations using point estimates and 95% CIs.

[^3] Marc Gastonguay. PAGE. 2004.

Thoughful considerations:

incorporate covariate effects in the model based on[^3]:

• Scientific and clinical interest.
• Mechanistic plausibility.
• Prior knowledge of covariate effect (e.g., body size on CL).
• Exploratory graphics (e.g., trend of parameter vs. covariate relationships)
• Avoid simultaneous inclusion of correlated/collinear covariates.

Exploratory graphics implying covariate-parameter relationships

Cautious evaluating corrleation/collinrarity (more important for FFEM)

library(tidyverse)
library(car)
library(here)
library(yspec)
library(GGally)

data <- readr::read_csv(
  here::here("wk04", "data", "pk.csv"),
  na = "."
)

# Spec
spec <- yspec::load_spec(
  here::here("wk04", "data", "pk.yml")
)

# Flags 
flags <- pull_meta(spec, "flags")

id <- data %>% distinct(ID, .keep_all = TRUE) %>% 
  ys_add_factors(spec,flags$edaCatCov,.suffix = "")

cov_all <- id %>% dplyr::select(DOSE, # dose was intented to be used as the outcome
                                flags$edaCatCov, 
                                flags$edaContCov) 

# names(cov_all)

# Examine correlation graphically
pmplots::pairs_plot(id, flags$edaContCov)

Cautious evaluating corrleation/collinrarity

pmplots::pm_grid(
  pmplots::wrap_cont_cat(id, x=flags$edaCatCov, y="WT") %>% 
    purrr::map(~ .x +pmplots::rot_x(angle=60)), 
  ncol = 3
  )

Variance inflation factor (VIF) for multicollinearity assessment

• A metric used in multiple regression to assess correlation among predictors (independent variables).
• Measures how much the variance of an estimated regression coefficient is increased due to collinearity.

For a covariate with 1 degree of freedom (\(df_i\)):

\[ VIF_i = \frac{1}{1-R_i^2} \]

For a general covariate with \(\geq1\) degree of freedom (\(df_i\))¹:

\[ GVIF_i = \frac{det(V_{jj})}{det(V_{jj}^*)} \\ aGVIF=GVIF^{1/(2\times df_i)} \]

• Provide a method to globally pre-screen covariate collinearity, include both continuous and categorical covariates, before NLME covariate modeling.

Variance inflation factor (VIF) criteria

\[ VIF_i = \frac{1}{1-R_i^2} \]

• VIF=1: no collinearity (\(R_i^2=0\))
• VIF=2: mild-moderate collinearity (\(R_i^2=0.5\))
• VIF=5: moderate-high collinearity (\(R_i^2=0.8\))
• VIF=10: high collinearity (\(R_i^2=0.9\))
• Usually target VIF<2, ideally VIF<1.5.

Variance inflation factor (VIF) implementations

# Fit full model
mod_full <- glm(DOSE ~ ., data = cov_all, family = "gaussian")

# Extract full model VIF
vif_full <- car::vif(mod_full)
vif_full

           GVIF Df GVIF^(1/(2*Df))
STUDY 11.366087  3        1.499461
RF    14.525495  3        1.562027
CP     7.695291  3        1.405090
AGE    1.081002  1        1.039713
WT     1.050855  1        1.025112
ALB    2.996708  1        1.731100
EGFR  10.638880  1        3.261730

# Fit reduced model
mod_reduce <- glm(DOSE ~ ., 
                  data = cov_all %>% dplyr::select(-ALB, # correlates with CP
                                                   -RF,  # correlates with EGFR
                                                   -STUDY), # correlates with multiple covariates
            family = "gaussian")

# Extract reduced model VIF
vif_reduce <- car::vif(mod_reduce)
vif_reduce

         GVIF Df GVIF^(1/(2*Df))
CP   1.065953  3        1.010702
AGE  1.024051  1        1.011954
WT   1.015808  1        1.007873
EGFR 1.049831  1        1.024612

Construct FFEM¹

\[ TVP = \theta_n \times \prod_{1}^{m} (\frac{COV_{mi}}{ref_m})^{\theta_(m+n)}\times \prod_{1}^{p}\theta_{p+m+n}^{COV_{pi}} \]

• \(TVP\): typical value of a model parameter.
• \(COV_{mi}\): \(m^{th}\) individual continuous covariate with \(ref_m\) as the reference value.
• \(COV_{pi}\): \(p^{th}\) individual categorical covariate (e.g., binary 0 or 1).
• \(\theta_n\): estimated parameter describing \(TVP\) for an individual with
  • \(COV_{mi} =ref_{m}\)
  • \(COV_{pi} = 0\)
• \(\theta_{m+n}\) and \(\theta_{p+m+n}\): estimated parameters describing the magnitude of the covariate-parameter relationship.

Construct FFEM example

\[ TVCL=THETA(1) \times (\frac{WT}{70})^{THETA(2)} \times (\frac{AGE}{43})^{THETA(3)} \times (\frac{CRCL}{80})^{THETA(4)} \]

• \(THETA(1)\) is the typical CL for a reference subject with 70kg WT, 43yo AGE with a CRCL of 80mL/min.
• If a covariate isn’t important, the power will be driven toward zero (null value) with high uncertainty (i.e., wide confidence interval).

FFEM Inference example: forest plot

Pros and Cons of selection methods¹

• Pros:
  • Widely used.
  • Easily performed
  • Freely available in open-source programs (e.g., PsN).
• Cons:
  • Statistical significance \(\neq\) clinical relevance.
  • Statistical insignificance \(\neq\) clinical irrelevance.
  • Potential for selection bias.
  • Selected covariates based on current data.
  • Problematic inference due to multiple testing (arbitrary P value cut-off).
  • Long run-times (i.e., computationally expansive).

Pros and Cons of Pre-defined methods¹